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			<h1 id="firstHeading" class="firstHeading">Low-pass filter</h1>
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				<p>A <b>low-pass filter</b> is a <a href="http://en.wikipedia.org/wiki/Filter_%28signal_processing%29" title="Filter (signal processing)">filter</a> that passes low-<a href="http://en.wikipedia.org/wiki/Frequency">frequency</a> <a href="http://en.wikipedia.org/wiki/Signal_%28electrical_engineering%29" title="Signal (electrical engineering)" class="mw-redirect">signals</a> but <a href="http://en.wikipedia.org/wiki/Attenuate" title="Attenuate" class="mw-redirect">attenuates</a> (reduces the <a href="http://en.wikipedia.org/wiki/Amplitude">amplitude</a> of) signals with frequencies higher than the <a href="http://en.wikipedia.org/wiki/Cutoff_frequency">cutoff frequency</a>. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a <b>high-cut filter</b>, or <b>treble cut filter</b> when used in audio applications. A low-pass filter is the opposite of a <a href="http://en.wikipedia.org/wiki/High-pass_filter">high-pass filter</a>, and a <a href="http://en.wikipedia.org/wiki/Band-pass_filter">band-pass filter</a> is a combination of a low-pass and a high-pass.</p>
<p>Low-pass filters exist in many different forms, including electronic circuits (such as a <i>hiss filter</i> used in <a href="http://en.wikipedia.org/wiki/Sound_recording" title="Sound recording" class="mw-redirect">audio</a>), <a href="http://en.wikipedia.org/wiki/Digital_filter" title="Digital filter">digital filters</a> for smoothing sets of data, acoustic barriers, blurring of images, and so on. The <a href="http://en.wikipedia.org/wiki/Moving_average_%28finance%29" title="Moving average (finance)" class="mw-redirect">moving average</a> operation used in fields such as finance is a particular kind of low-pass filter, and can be analyzed with the same <a href="http://en.wikipedia.org/wiki/Signal_processing">signal processing</a>
 techniques as are used for other low-pass filters. Low-pass filters 
provide a smoother form of a signal, removing the short-term 
fluctuations, and leaving the longer-term trend.</p>
<table id="toc" class="toc">
<tbody><tr>
<td>
<div id="toctitle">
<h2>Contents</h2>
 <span class="toctoggle">[<a href="#" class="internal" id="togglelink">hide</a>]</span></div>
<ul>
<li class="toclevel-1 tocsection-1"><a href="#Examples_of_low-pass_filters"><span class="tocnumber">1</span> <span class="toctext">Examples of low-pass filters</span></a>
<ul>
<li class="toclevel-2 tocsection-2"><a href="#Acoustic"><span class="tocnumber">1.1</span> <span class="toctext">Acoustic</span></a></li>
<li class="toclevel-2 tocsection-3"><a href="#Electronic"><span class="tocnumber">1.2</span> <span class="toctext">Electronic</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-4"><a href="#Ideal_and_real_filters"><span class="tocnumber">2</span> <span class="toctext">Ideal and real filters</span></a></li>
<li class="toclevel-1 tocsection-5"><a href="#Continuous-time_low-pass_filters"><span class="tocnumber">3</span> <span class="toctext">Continuous-time low-pass filters</span></a>
<ul>
<li class="toclevel-2 tocsection-6"><a href="#Laplace_notation"><span class="tocnumber">3.1</span> <span class="toctext">Laplace notation</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-7"><a href="#Electronic_low-pass_filters"><span class="tocnumber">4</span> <span class="toctext">Electronic low-pass filters</span></a>
<ul>
<li class="toclevel-2 tocsection-8"><a href="#Passive_electronic_realization"><span class="tocnumber">4.1</span> <span class="toctext">Passive electronic realization</span></a></li>
<li class="toclevel-2 tocsection-9"><a href="#Active_electronic_realization"><span class="tocnumber">4.2</span> <span class="toctext">Active electronic realization</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-10"><a href="#Discrete-time_realization"><span class="tocnumber">5</span> <span class="toctext">Discrete-time realization</span></a>
<ul>
<li class="toclevel-2 tocsection-11"><a href="#Algorithmic_implementation"><span class="tocnumber">5.1</span> <span class="toctext">Algorithmic implementation</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-12"><a href="#See_also"><span class="tocnumber">6</span> <span class="toctext">See also</span></a></li>
<li class="toclevel-1 tocsection-13"><a href="#References"><span class="tocnumber">7</span> <span class="toctext">References</span></a></li>
<li class="toclevel-1 tocsection-14"><a href="#External_links"><span class="tocnumber">8</span> <span class="toctext">External links</span></a></li>
</ul>
</td>
</tr>
</tbody></table>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Low-pass_filter&amp;action=edit&amp;section=1" title="Edit section: Examples of low-pass filters">edit</a>]</span> <span class="mw-headline" id="Examples_of_low-pass_filters">Examples of low-pass filters</span></h2>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Low-pass_filter&amp;action=edit&amp;section=2" title="Edit section: Acoustic">edit</a>]</span> <span class="mw-headline" id="Acoustic">Acoustic</span></h3>
<p>A stiff physical barrier tends to reflect higher sound frequencies, 
and so acts as a low-pass filter for transmitting sound. When music is 
playing in another room, the low notes are easily heard, while the high 
notes are attenuated.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Low-pass_filter&amp;action=edit&amp;section=3" title="Edit section: Electronic">edit</a>]</span> <span class="mw-headline" id="Electronic">Electronic</span></h3>
<p>In an electronic low-pass <a href="http://en.wikipedia.org/wiki/RC_filter" class="mw-redirect" title="RC filter">RC filter</a>
 for voltage signals, high frequencies contained in the input signal are
 attenuated but the filter has little attenuation below its <a href="http://en.wikipedia.org/wiki/Cutoff_frequency">cutoff frequency</a> which is determined by its <a href="http://en.wikipedia.org/wiki/RC_time_constant">RC time constant</a>.</p>
<p>For current signals, a similar circuit using a resistor and capacitor in <a href="http://en.wikipedia.org/wiki/Parallel_circuits#Parallel_circuits" title="Parallel circuits" class="mw-redirect">parallel</a> works in a similar manner. See <a href="http://en.wikipedia.org/wiki/Current_divider">current divider</a> discussed in more detail <a href="http://en.wikipedia.org/wiki/Low-pass_filter#Electronic_low-pass_filters" title="Low-pass filter">below</a>.</p>
<p>Electronic low-pass filters are used to drive <a href="http://en.wikipedia.org/wiki/Subwoofer" title="Subwoofer">subwoofers</a> and other types of <a href="http://en.wikipedia.org/wiki/Loudspeaker" title="Loudspeaker">loudspeakers</a>, to block high pitches that they can't efficiently broadcast.</p>
<p>Radio transmitters use low-pass filters to block <a href="http://en.wikipedia.org/wiki/Harmonic">harmonic</a> emissions which might cause interference with other communications.</p>
<p>The tone knob found on many <a href="http://en.wikipedia.org/wiki/Electric_guitars" class="mw-redirect" title="Electric guitars">electric guitars</a> is a low-pass filter used to reduce the amount of treble in the sound.</p>
<p>An <a href="http://en.wikipedia.org/wiki/Integrator">integrator</a> is another example of a single <a href="http://en.wikipedia.org/wiki/Time_constant#Time_constants_in_electrical_circuits">time constant</a> low-pass filter.<sup id="cite_ref-0" class="reference"><a href="#cite_note-0"><span>[</span>1<span>]</span></a></sup></p>
<p>Telephone lines fitted with <a href="http://en.wikipedia.org/wiki/DSL_splitter" title="DSL splitter" class="mw-redirect">DSL splitters</a> use low-pass and <a href="http://en.wikipedia.org/wiki/High-pass_filter" title="High-pass filter">high-pass</a> filters to separate <a href="http://en.wikipedia.org/wiki/Digital_Subscriber_Line" title="Digital Subscriber Line">DSL</a> and <a href="http://en.wikipedia.org/wiki/Plain_old_telephone_service" title="Plain old telephone service">POTS</a> signals sharing the same <a href="http://en.wikipedia.org/wiki/Twisted_pair" title="Twisted pair">pair</a> of wires.</p>
<p>Low-pass filters also play a significant role in the sculpting of sound for <a href="http://en.wikipedia.org/wiki/Electronic_music">electronic music</a> as created by analogue <a href="http://en.wikipedia.org/wiki/Synthesiser" title="Synthesiser" class="mw-redirect">synthesisers</a>. <i>See <a href="http://en.wikipedia.org/wiki/Subtractive_synthesis">subtractive synthesis</a>.</i></p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Low-pass_filter&amp;action=edit&amp;section=4" title="Edit section: Ideal and real filters">edit</a>]</span> <span class="mw-headline" id="Ideal_and_real_filters">Ideal and real filters</span></h2>
<div class="thumb tright">
<div class="thumbinner" style="width:222px;"><a href="http://en.wikipedia.org/wiki/File:Sinc_function_%28normalized%29.svg" class="image"><img alt="" src="wikipedia-Low_pass_filter_pliki/220px-Sinc_function_normalized.png" class="thumbimage" height="151" width="220"></a>
<div class="thumbcaption">
<div class="magnify"><a href="http://en.wikipedia.org/wiki/File:Sinc_function_%28normalized%29.svg" class="internal" title="Enlarge"><img src="wikipedia-Low_pass_filter_pliki/magnify-clip.png" alt="" height="11" width="15"></a></div>
The <a href="http://en.wikipedia.org/wiki/Sinc_function">sinc function</a>, the <a href="http://en.wikipedia.org/wiki/Impulse_response">impulse response</a> of an ideal low-pass filter.</div>
</div>
</div>
<p>An <a href="http://en.wikipedia.org/wiki/Sinc_filter" title="Sinc filter">ideal low-pass filter</a> completely eliminates all frequencies above the <a href="http://en.wikipedia.org/wiki/Cutoff_frequency">cutoff frequency</a> while passing those below unchanged: its <a href="http://en.wikipedia.org/wiki/Frequency_response">frequency response</a> is a <a href="http://en.wikipedia.org/wiki/Rectangular_function">rectangular function</a>, and is a <a href="http://en.wikipedia.org/wiki/Brick-wall_filter" class="mw-redirect" title="Brick-wall filter">brick-wall filter</a>.
 The transition region present in practical filters does not exist in an
 ideal filter. An ideal low-pass filter can be realized mathematically 
(theoretically) by multiplying a signal by the rectangular function in 
the frequency domain or, equivalently, <a href="http://en.wikipedia.org/wiki/Convolution">convolution</a> with its <a href="http://en.wikipedia.org/wiki/Impulse_response">impulse response</a>, a <a href="http://en.wikipedia.org/wiki/Sinc_function">sinc function</a>, in the time domain.</p>
<p>However, the ideal filter is impossible to realize without also 
having signals of infinite extent in time, and so generally needs to be 
approximated for real ongoing signals, because the sinc function's 
support region extends to all past and future times. The filter would 
therefore need to have infinite delay, or knowledge of the infinite 
future and past, in order to perform the convolution. It is effectively 
realizable for pre-recorded digital signals by assuming extensions of 
zero into the past and future, or more typically by making the signal 
repetitive and using Fourier analysis.</p>
<p>Real filters for <a href="http://en.wikipedia.org/wiki/Real-time_computing" title="Real-time computing">real-time</a> applications approximate the ideal filter by truncating and <a href="http://en.wikipedia.org/wiki/Window_function" title="Window function">windowing</a> the infinite impulse response to make a <a href="http://en.wikipedia.org/wiki/Finite_impulse_response">finite impulse response</a>;
 applying that filter requires delaying the signal for a moderate period
 of time, allowing the computation to "see" a little bit into the 
future. This delay is manifested as <a href="http://en.wikipedia.org/wiki/Phase_%28waves%29" title="Phase (waves)">phase shift</a>. Greater accuracy in approximation requires a longer delay.</p>
<p>An ideal low-pass filter results in <a href="http://en.wikipedia.org/wiki/Ringing_artifacts">ringing artifacts</a> via the <a href="http://en.wikipedia.org/wiki/Gibbs_phenomenon">Gibbs phenomenon</a>. These can be reduced or worsened by choice of windowing function, and the <a href="http://en.wikipedia.org/wiki/Window_function#Filter_design" title="Window function">design and choice of real filters</a>
 involves understanding and minimizing these artifacts. For example, 
"simple truncation [of sinc] causes severe ringing artifacts," in signal
 reconstruction, and to reduce these artifacts one uses window functions
 "which drop off more smoothly at the edges."<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span>[</span>2<span>]</span></a></sup></p>
<p>The <a href="http://en.wikipedia.org/wiki/Whittaker%E2%80%93Shannon_interpolation_formula">Whittaker–Shannon interpolation formula</a> describes how to use a perfect low-pass filter to reconstruct a <a href="http://en.wikipedia.org/wiki/Continuous_signal">continuous signal</a> from a sampled <a href="http://en.wikipedia.org/wiki/Digital_signal">digital signal</a>. Real <a href="http://en.wikipedia.org/wiki/Digital-to-analog_converter" title="Digital-to-analog converter">digital-to-analog converters</a> use real filter approximations.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Low-pass_filter&amp;action=edit&amp;section=5" title="Edit section: Continuous-time low-pass filters">edit</a>]</span> <span class="mw-headline" id="Continuous-time_low-pass_filters">Continuous-time low-pass filters</span></h2>
<div class="thumb tright">
<div class="thumbinner" style="width:352px;"><a href="http://en.wikipedia.org/wiki/File:Butterworth_response.svg" class="image"><img alt="" src="wikipedia-Low_pass_filter_pliki/350px-Butterworth_response.png" class="thumbimage" height="263" width="350"></a>
<div class="thumbcaption">
<div class="magnify"><a href="http://en.wikipedia.org/wiki/File:Butterworth_response.svg" class="internal" title="Enlarge"><img src="wikipedia-Low_pass_filter_pliki/magnify-clip.png" alt="" height="11" width="15"></a></div>
The gain-magnitude frequency response of a first-order (one-pole) low-pass filter. <i>Power gain</i> is shown in decibels (i.e., a 3 <a href="http://en.wikipedia.org/wiki/Decibel" title="Decibel">dB</a> decline reflects an additional half-power attenuation). <a href="http://en.wikipedia.org/wiki/Angular_frequency">Angular frequency</a> is shown on a logarithmic scale in units of radians per second.</div>
</div>
</div>
<p>There are many different types of filter circuits, with different 
responses to changing frequency. The frequency response of a filter is 
generally represented using a <a href="http://en.wikipedia.org/wiki/Bode_plot">Bode plot</a>, and the filter is characterized by its <a href="http://en.wikipedia.org/wiki/Cutoff_frequency">cutoff frequency</a> and rate of frequency <a href="http://en.wikipedia.org/wiki/Roll-off" title="Roll-off">rolloff</a>. In all cases, at the <i>cutoff frequency,</i> the filter <a href="http://en.wikipedia.org/wiki/Attenuate" title="Attenuate" class="mw-redirect">attenuates</a> the input power by half or 3 dB. So the <b>order</b> of the filter determines the amount of additional attenuation for frequencies higher than the cutoff frequency.</p>
<ul>
<li>A <b>first-order filter</b>, for example, will reduce the signal amplitude by half (so power reduces by a factor of 4), or <span style="white-space:nowrap;">6 dB</span>, every time the frequency doubles (goes up one <a href="http://en.wikipedia.org/wiki/Octave">octave</a>);
 more precisely, the power rolloff approaches 20 dB per decade in the 
limit of high frequency. The magnitude Bode plot for a first-order 
filter looks like a horizontal line below the <a href="http://en.wikipedia.org/wiki/Cutoff_frequency">cutoff frequency</a>,
 and a diagonal line above the cutoff frequency. There is also a "knee 
curve" at the boundary between the two, which smoothly transitions 
between the two straight line regions. If the <a href="http://en.wikipedia.org/wiki/Transfer_function">transfer function</a> of a first-order low-pass filter has a <a href="http://en.wikipedia.org/wiki/Zero_%28complex_analysis%29" title="Zero (complex analysis)">zero</a> as well as a <a href="http://en.wikipedia.org/wiki/Pole_%28complex_analysis%29" title="Pole (complex analysis)">pole</a>,
 the Bode plot will flatten out again, at some maximum attenuation of 
high frequencies; such an effect is caused for example by a little bit 
of the input leaking around the one-pole filter; this one-pole–one-zero 
filter is still a first-order low-pass. <i>See <a href="http://en.wikipedia.org/wiki/Pole%E2%80%93zero_plot">Pole–zero plot</a> and <a href="http://en.wikipedia.org/wiki/RC_circuit">RC circuit</a>.</i></li>
</ul>
<ul>
<li>A <b>second-order filter</b> attenuates higher frequencies more 
steeply. The Bode plot for this type of filter resembles that of a 
first-order filter, except that it falls off more quickly. For example, a
 second-order <a href="http://en.wikipedia.org/wiki/Butterworth_filter">Butterworth filter</a>
 will reduce the signal amplitude to one fourth its original level every
 time the frequency doubles (so power decreases by 12 dB per octave, or 
40 dB per decade). Other all-pole second-order filters may roll off at 
different rates initially depending on their <a href="http://en.wikipedia.org/wiki/Q_factor">Q factor</a>,
 but approach the same final rate of 12 dB per octave; as with the 
first-order filters, zeroes in the transfer function can change the 
high-frequency asymptote. See <a href="http://en.wikipedia.org/wiki/RLC_circuit">RLC circuit</a>.</li>
</ul>
<ul>
<li>Third- and higher-order filters are defined similarly. In general, the final rate of power rolloff for an order-<span class="texhtml"><i>n</i></span> all-pole filter is <span class="texhtml">6<i>n</i></span> dB per octave (i.e., <span class="texhtml">20<i>n</i></span> dB per decade).</li>
</ul>
<p>On any Butterworth filter, if one extends the horizontal line to the right and the diagonal line to the upper-left (the <a href="http://en.wikipedia.org/wiki/Asymptote" title="Asymptote">asymptotes</a>
 of the function), they will intersect at exactly the "cutoff 
frequency". The frequency response at the cutoff frequency in a 
first-order filter is 3 dB below the horizontal line. The various types 
of filters&nbsp;– <a href="http://en.wikipedia.org/wiki/Butterworth_filter">Butterworth filter</a>, <a href="http://en.wikipedia.org/wiki/Chebyshev_filter">Chebyshev filter</a>, <a href="http://en.wikipedia.org/wiki/Bessel_filter">Bessel filter</a>, etc.&nbsp;– all have different-looking "knee curves". Many second-order filters are designed to have "peaking" or <a href="http://en.wikipedia.org/wiki/Electrical_resonance" title="Electrical resonance">resonance</a>, causing their frequency response at the cutoff frequency to be <i>above</i> the horizontal line. <i>See <a href="http://en.wikipedia.org/wiki/Electronic_filter">electronic filter</a> for other types.</i></p>
<p>The meanings of 'low' and 'high'&nbsp;– that is, the <a href="http://en.wikipedia.org/wiki/Cutoff_frequency">cutoff frequency</a>&nbsp;–
 depend on the characteristics of the filter. The term "low-pass filter"
 merely refers to the shape of the filter's response; a high-pass filter
 could be built that cuts off at a lower frequency than any low-pass 
filter&nbsp;– it is their responses that set them apart. Electronic 
circuits can be devised for any desired frequency range, right up 
through microwave frequencies (above 1&nbsp;GHz) and higher.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Low-pass_filter&amp;action=edit&amp;section=6" title="Edit section: Laplace notation">edit</a>]</span> <span class="mw-headline" id="Laplace_notation">Laplace notation</span></h3>
<p>Continuous-time filters can also be described in terms of the <a href="http://en.wikipedia.org/wiki/Laplace_transform">Laplace transform</a> of their <a href="http://en.wikipedia.org/wiki/Impulse_response">impulse response</a>
 in a way that allows all of the characteristics of the filter to be 
easily analyzed by considering the pattern of poles and zeros of the 
Laplace transform in the complex plane (in discrete time, one can 
similarly consider the <a href="http://en.wikipedia.org/wiki/Z-transform">Z-transform</a> of the impulse response).</p>
<p>For example, a first-order low-pass filter can be described in Laplace notation as</p>
<dl>
<dd><img class="tex" alt="
\frac{\text{Output}}{\text{Input}} = K \frac{1}{1 + s \tau}
" src="wikipedia-Low_pass_filter_pliki/e54d23f65493a2db37b7a715643aea15.png"></dd>
</dl>
<p>where <i>s</i> is the Laplace transform variable, <i>τ</i> is the filter <a href="http://en.wikipedia.org/wiki/Time_constant">time constant</a>, and <i>K</i> is the filter <a href="http://en.wikipedia.org/wiki/Passband">passband</a> <a href="http://en.wikipedia.org/wiki/Gain">gain</a>.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Low-pass_filter&amp;action=edit&amp;section=7" title="Edit section: Electronic low-pass filters">edit</a>]</span> <span class="mw-headline" id="Electronic_low-pass_filters">Electronic low-pass filters</span></h2>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Low-pass_filter&amp;action=edit&amp;section=8" title="Edit section: Passive electronic realization">edit</a>]</span> <span class="mw-headline" id="Passive_electronic_realization">Passive electronic realization</span></h3>
<div class="thumb tright">
<div class="thumbinner" style="width:202px;"><a href="http://en.wikipedia.org/wiki/File:RC_Divider.svg" class="image"><img alt="" src="wikipedia-Low_pass_filter_pliki/200px-RC_Divider.png" class="thumbimage" height="187" width="200"></a>
<div class="thumbcaption">
<div class="magnify"><a href="http://en.wikipedia.org/wiki/File:RC_Divider.svg" class="internal" title="Enlarge"><img src="wikipedia-Low_pass_filter_pliki/magnify-clip.png" alt="" height="11" width="15"></a></div>
Passive, first order low-pass RC filter</div>
</div>
</div>
<p>One simple <a href="http://en.wikipedia.org/wiki/Electrical_circuit" class="mw-redirect" title="Electrical circuit">electrical circuit</a> that will serve as a low-pass filter consists of a <a href="http://en.wikipedia.org/wiki/Resistor">resistor</a> in series with a <a href="http://en.wikipedia.org/wiki/External_electric_load" title="External electric load" class="mw-redirect">load</a>, and a <a href="http://en.wikipedia.org/wiki/Capacitor">capacitor</a> in parallel with the load. The capacitor exhibits <a href="http://en.wikipedia.org/wiki/Reactance_%28electronics%29" title="Reactance (electronics)" class="mw-redirect">reactance</a>,
 and blocks low-frequency signals, causing them to go through the load 
instead. At higher frequencies the reactance drops, and the capacitor 
effectively functions as a short circuit. The combination of resistance 
and capacitance gives you the <a href="http://en.wikipedia.org/wiki/Time_constant">time constant</a> of the filter <span class="texhtml">τ = <i>R</i><i>C</i></span> (represented by the Greek letter <a href="http://en.wikipedia.org/wiki/Tau">tau</a>). The break frequency, also called the turnover frequency or <a href="http://en.wikipedia.org/wiki/Cutoff_frequency">cutoff frequency</a> (in hertz), is determined by the time constant:</p>
<dl>
<dd><img class="tex" alt="
f_\mathrm{c} = {1 \over 2 \pi \tau } = {1 \over 2 \pi R C}
" src="wikipedia-Low_pass_filter_pliki/143e8b2e376f030107ce81c7efa56d26.png"></dd>
</dl>
<p>or equivalently (in <a href="http://en.wikipedia.org/wiki/Radians" class="mw-redirect" title="Radians">radians</a> per second):</p>
<dl>
<dd><img class="tex" alt="
\omega_\mathrm{c} = {1 \over \tau} = { 1 \over R C}.
" src="wikipedia-Low_pass_filter_pliki/a3e1af4a0dcad641af41d3f86cd11548.png"></dd>
</dl>
<p>One way to understand this circuit is to focus on the time the 
capacitor takes to charge. It takes time to charge or discharge the 
capacitor through that resistor:</p>
<ul>
<li>At low frequencies, there is plenty of time for the capacitor to charge up to practically the same voltage as the input voltage.</li>
<li>At high frequencies, the capacitor only has time to charge up a 
small amount before the input switches direction. The output goes up and
 down only a small fraction of the amount the input goes up and down. At
 double the frequency, there's only time for it to charge up half the 
amount.</li>
</ul>
<p>Another way to understand this circuit is with the idea of <a href="http://en.wikipedia.org/wiki/Reactance_%28electronics%29" title="Reactance (electronics)" class="mw-redirect">reactance</a> at a particular frequency:</p>
<ul>
<li>Since <a href="http://en.wikipedia.org/wiki/Direct_current" title="Direct current">DC</a> cannot flow through the capacitor, DC input must "flow out" the path marked <span class="texhtml"><i>V</i><sub>out</sub></span> (analogous to removing the capacitor).</li>
<li>Since <a href="http://en.wikipedia.org/wiki/Alternating_current" title="Alternating current">AC</a>
 flows very well through the capacitor — almost as well as it flows 
through solid wire — AC input "flows out" through the capacitor, 
effectively <a href="http://en.wikipedia.org/wiki/Short_circuit" title="Short circuit">short circuiting</a> to ground (analogous to replacing the capacitor with just a wire).</li>
</ul>
<p>The capacitor is not an "on/off" object (like the block or pass 
fluidic explanation above). The capacitor will variably act between 
these two extremes. It is the <a href="http://en.wikipedia.org/wiki/Bode_plot">Bode plot</a> and <a href="http://en.wikipedia.org/wiki/Frequency_response">frequency response</a> that show this variability.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Low-pass_filter&amp;action=edit&amp;section=9" title="Edit section: Active electronic realization">edit</a>]</span> <span class="mw-headline" id="Active_electronic_realization">Active electronic realization</span></h3>
<div class="thumb tright">
<div class="thumbinner" style="width:302px;"><a href="http://en.wikipedia.org/wiki/File:Active_Lowpass_Filter_RC.svg" class="image"><img alt="" src="wikipedia-Low_pass_filter_pliki/300px-Active_Lowpass_Filter_RC.png" class="thumbimage" height="208" width="300"></a>
<div class="thumbcaption">
<div class="magnify"><a href="http://en.wikipedia.org/wiki/File:Active_Lowpass_Filter_RC.svg" class="internal" title="Enlarge"><img src="wikipedia-Low_pass_filter_pliki/magnify-clip.png" alt="" height="11" width="15"></a></div>
An active low-pass filter</div>
</div>
</div>
<p>Another type of electrical circuit is an <i>active</i> low-pass filter.</p>
<p>In the <a href="http://en.wikipedia.org/wiki/Operational_amplifier">operational amplifier</a> circuit shown in the figure, the cutoff frequency (in <a href="http://en.wikipedia.org/wiki/Hertz">hertz</a>) is defined as:</p>
<dl>
<dd><img class="tex" alt="f_{\text{c}} = \frac{1}{2 \pi R_2 C}" src="wikipedia-Low_pass_filter_pliki/7e3a67cae3881e5db385ff32801cb763.png"></dd>
</dl>
<p>or equivalently (in radians per second):</p>
<dl>
<dd><img class="tex" alt="\omega_{\text{c}} = \frac{1}{R_2 C}." src="wikipedia-Low_pass_filter_pliki/dfe4fc50399eea6ca35462f7406571ab.png"></dd>
</dl>
<p>The gain in the passband is −<i>R</i><sub>2</sub>/<i>R</i><sub>1</sub>, and the <a href="http://en.wikipedia.org/wiki/Stopband">stopband</a> drops off at −6&nbsp;dB per octave as it is a first-order filter.</p>
<p>Sometimes, a simple gain amplifier (as opposed to the very-high-gain 
operational amplifier) is turned into a low-pass filter by simply adding
 a feedback capacitor <i>C</i>. This feedback decreases the frequency response at high frequencies via the <a href="http://en.wikipedia.org/wiki/Miller_effect">Miller effect</a>,
 and helps to avoid oscillation in the amplifier. For example, an audio 
amplifier can be made into a low-pass filter with cutoff frequency 
100&nbsp;kHz to reduce gain at frequencies which would otherwise 
oscillate. Since the audio band (what we can hear) only goes up to 
20&nbsp;kHz or so, the frequencies of interest fall entirely in the <a href="http://en.wikipedia.org/wiki/Passband">passband</a>, and the amplifier behaves the same way as far as audio is concerned.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Low-pass_filter&amp;action=edit&amp;section=10" title="Edit section: Discrete-time realization">edit</a>]</span> <span class="mw-headline" id="Discrete-time_realization">Discrete-time realization</span></h2>
<div class="dablink">For another method of conversion from continuous- to discrete-time, see <a href="http://en.wikipedia.org/wiki/Bilinear_transform">Bilinear transform</a>.</div>
<p>The effect of a low-pass filter can be simulated on a computer by analyzing its behavior in the time domain, and then <a href="http://en.wikipedia.org/wiki/Discrete_signal" title="Discrete signal">discretizing</a> the model.</p>
<div class="thumb tright">
<div class="thumbinner" style="width:252px;"><a href="http://en.wikipedia.org/wiki/File:1st_Order_Lowpass_Filter_RC.svg" class="image"><img alt="" src="wikipedia-Low_pass_filter_pliki/250px-1st_Order_Lowpass_Filter_RC.png" class="thumbimage" height="140" width="250"></a>
<div class="thumbcaption">A simple low-pass <a href="http://en.wikipedia.org/wiki/RC_circuit" title="RC circuit">RC filter</a></div>
</div>
</div>
<p>From the circuit diagram to the right, according to <a href="http://en.wikipedia.org/wiki/Kirchhoff%27s_circuit_laws" title="Kirchhoff's circuit laws">Kirchoff's Laws</a> and the definition of <a href="http://en.wikipedia.org/wiki/Capacitance">capacitance</a>:</p>
<dl>
<dd>
<table style="border-collapse: collapse; background: none repeat scroll 0% 0% transparent; margin: 0pt; border: medium none;">
<tbody><tr>
<td style="vertical-align: middle; border: medium none; padding: 0.08em;" nowrap="nowrap">
<p style="margin:0;"><img class="tex" alt="v_{\text{in}}(t) - v_{\text{out}}(t) = R \; i(t)" src="wikipedia-Low_pass_filter_pliki/c80c1989dbec78a5f268d25bb3cc9502.png"></p>
</td>
<td style="vertical-align: middle; width: 99%; border: medium none; padding: 0.08em;">
<p style="margin:0;"></p>
<table style="border-collapse: collapse; background: none repeat scroll 0% 0% transparent; margin: 0pt; border: medium none; width: 99%;">
<tbody><tr>
<td style="border: medium none; padding: 0.08em;" rowspan="2">
<p style="margin:0; font-size:4pt;">&nbsp;</p>
</td>
<td style="width: 100%; border: medium none; padding: 0.08em;">
<p style="margin:0; font-size:1pt;">&nbsp;</p>
</td>
<td style="border: medium none; padding: 0.08em;" rowspan="2">
<p style="margin:0; font-size:4pt;">&nbsp;</p>
</td>
</tr>
<tr>
<td style="border-left: medium none; border-width: 0px medium medium; border-style: none; border-color: rgb(229, 229, 229) -moz-use-text-color -moz-use-text-color; padding: 0.08em;">
<p style="margin:0; font-size:1pt;">&nbsp;</p>
</td>
</tr>
</tbody></table>
</td>
<td style="vertical-align: middle; border: medium none; padding: 0.08em;" nowrap="nowrap">
<p style="margin:0pt;"><b>(<cite id="math_V"></cite><span class="reference plainlinksneverexpand"><cite id="math_V"><a href="#equation_V">V</a></cite><b><cite id="math_V"></cite>)</b></span></b></p>
</td>
</tr>
</tbody></table>
</dd>
</dl>
<dl>
<dd>
<table style="border-collapse: collapse; background: none repeat scroll 0% 0% transparent; margin: 0pt; border: medium none;">
<tbody><tr>
<td style="vertical-align: middle; border: medium none; padding: 0.08em;" nowrap="nowrap">
<p style="margin:0;"><img class="tex" alt="Q_c(t) = C \, v_{\text{out}}(t)" src="wikipedia-Low_pass_filter_pliki/06fa1586bdfbfeb9ac8c0c7e17addebc.png"></p>
</td>
<td style="vertical-align: middle; width: 99%; border: medium none; padding: 0.08em;">
<p style="margin:0;"></p>
<table style="border-collapse: collapse; background: none repeat scroll 0% 0% transparent; margin: 0pt; border: medium none; width: 99%;">
<tbody><tr>
<td style="border: medium none; padding: 0.08em;" rowspan="2">
<p style="margin:0; font-size:4pt;">&nbsp;</p>
</td>
<td style="width: 100%; border: medium none; padding: 0.08em;">
<p style="margin:0; font-size:1pt;">&nbsp;</p>
</td>
<td style="border: medium none; padding: 0.08em;" rowspan="2">
<p style="margin:0; font-size:4pt;">&nbsp;</p>
</td>
</tr>
<tr>
<td style="border-left: medium none; border-width: 0px medium medium; border-style: none; border-color: rgb(229, 229, 229) -moz-use-text-color -moz-use-text-color; padding: 0.08em;">
<p style="margin:0; font-size:1pt;">&nbsp;</p>
</td>
</tr>
</tbody></table>
</td>
<td style="vertical-align: middle; border: medium none; padding: 0.08em;" nowrap="nowrap">
<p style="margin:0pt;"><b>(<cite id="math_Q"></cite><span class="reference plainlinksneverexpand"><cite id="math_Q"><a href="#equation_Q">Q</a></cite><b><cite id="math_Q"></cite>)</b></span></b></p>
</td>
</tr>
</tbody></table>
</dd>
</dl>
<dl>
<dd>
<table style="border-collapse: collapse; background: none repeat scroll 0% 0% transparent; margin: 0pt; border: medium none;">
<tbody><tr>
<td style="vertical-align: middle; border: medium none; padding: 0.08em;" nowrap="nowrap">
<p style="margin:0;"><img class="tex" alt="i(t) = \frac{\operatorname{d} Q_c}{\operatorname{d} t} \, ," src="wikipedia-Low_pass_filter_pliki/115518e4e52fe629a9b519e50d594ccb.png"></p>
</td>
<td style="vertical-align: middle; width: 99%; border: medium none; padding: 0.08em;">
<p style="margin:0;"></p>
<table style="border-collapse: collapse; background: none repeat scroll 0% 0% transparent; margin: 0pt; border: medium none; width: 99%;">
<tbody><tr>
<td style="border: medium none; padding: 0.08em;" rowspan="2">
<p style="margin:0; font-size:4pt;">&nbsp;</p>
</td>
<td style="width: 100%; border: medium none; padding: 0.08em;">
<p style="margin:0; font-size:1pt;">&nbsp;</p>
</td>
<td style="border: medium none; padding: 0.08em;" rowspan="2">
<p style="margin:0; font-size:4pt;">&nbsp;</p>
</td>
</tr>
<tr>
<td style="border-left: medium none; border-width: 0px medium medium; border-style: none; border-color: rgb(229, 229, 229) -moz-use-text-color -moz-use-text-color; padding: 0.08em;">
<p style="margin:0; font-size:1pt;">&nbsp;</p>
</td>
</tr>
</tbody></table>
</td>
<td style="vertical-align: middle; border: medium none; padding: 0.08em;" nowrap="nowrap">
<p style="margin:0pt;"><b>(<cite id="math_I"></cite><span class="reference plainlinksneverexpand"><cite id="math_I"><a href="#equation_I">I</a></cite><b><cite id="math_I"></cite>)</b></span></b></p>
</td>
</tr>
</tbody></table>
</dd>
</dl>
<p>where <span class="texhtml"><i>Q</i><sub><i>c</i></sub>(<i>t</i>)</span> is the charge stored in the capacitor at time <span class="texhtml"><i>t</i></span>. Substituting equation <cite id="equation_Q" style="font-style: normal;"><b><a href="#math_Q">Q</a></b></cite> into equation <cite id="equation_I" style="font-style: normal;"><b><a href="#math_I">I</a></b></cite> gives <img class="tex" alt="i(t) = C \frac{\operatorname{d}v_{\text{out}}}{\operatorname{d}t}" src="wikipedia-Low_pass_filter_pliki/2ef2ca74811215361a79d0d7d079eba4.png">, which can be substituted into equation <cite id="equation_V" style="font-style: normal;"><b><a href="#math_V">V</a></b></cite> so that:</p>
<dl>
<dd><img class="tex" alt="v_{\text{in}}(t) - v_{\text{out}}(t) = RC \frac{\operatorname{d}v_{\text{out}}}{\operatorname{d}t}.\," src="wikipedia-Low_pass_filter_pliki/c01b5bbafe3c41b6845d0025eb5627a9.png"></dd>
</dl>
<p>This equation can be discretized. For simplicity, assume that samples
 of the input and output are taken at evenly-spaced points in time 
separated by <span class="texhtml">Δ<sub><i>T</i></sub></span> time. Let the samples of <span class="texhtml"><i>v</i><sub>in</sub></span> be represented by the sequence <img class="tex" alt="(x_1, x_2, \ldots, x_n)" src="wikipedia-Low_pass_filter_pliki/8b98f712c02e31f319616f5b8bcc05fa.png">, and let <span class="texhtml"><i>v</i><sub>out</sub></span> be represented by the sequence <img class="tex" alt="(y_1, y_2, \ldots, y_n)" src="wikipedia-Low_pass_filter_pliki/f0c206e68a41fd151c5671d9772e2dfc.png"> which correspond to the same points in time. Making these substitutions:</p>
<dl>
<dd><img class="tex" alt="x_i - y_i = RC \, \frac{y_{i}-y_{i-1}}{\Delta_T}.\," src="wikipedia-Low_pass_filter_pliki/80119fba0910d7225f72b8113cba748e.png"></dd>
</dl>
<p>And rearranging terms gives the <a href="http://en.wikipedia.org/wiki/Recurrence_relation">recurrence relation</a></p>
<dl>
<dd><img class="tex" alt="y_i = \overbrace{x_i \left( \frac{\Delta_T}{RC + \Delta_T} \right)}^{\text{Input contribution}} + \overbrace{y_{i-1} \left( \frac{RC}{RC + \Delta_T} \right)}^{\text{Inertia from previous output}}." src="wikipedia-Low_pass_filter_pliki/968b4de0f6e7b6623a174db7c6080880.png"></dd>
</dl>
<p>That is, this discrete-time implementation of a simple RC low-pass filter is the <a href="http://en.wikipedia.org/wiki/Exponential_smoothing" title="Exponential smoothing">exponentially-weighted moving average</a></p>
<dl>
<dd><img class="tex" alt="y_i = \alpha x_i + (1 - \alpha) y_{i-1} \qquad \text{where} \qquad \alpha \triangleq \frac{\Delta_T}{RC + \Delta_T}.\," src="wikipedia-Low_pass_filter_pliki/bef174f75da549ab42bdc930224fcfa8.png"></dd>
</dl>
<p>By definition, the <i>smoothing factor</i> <img class="tex" alt="0 \leq \alpha \leq 1" src="wikipedia-Low_pass_filter_pliki/1e7fffc2c1cba34188529b70396467a1.png">. The expression for <span class="texhtml">α</span> yields the equivalent <a href="http://en.wikipedia.org/wiki/Time_constant">time constant</a> <span class="texhtml"><i>R</i><i>C</i></span> in terms of the sampling period <span class="texhtml">Δ<sub><i>T</i></sub></span> and smoothing factor <span class="texhtml">α</span>:</p>
<dl>
<dd><img class="tex" alt="RC = \Delta_T \left( \frac{1 - \alpha}{\alpha} \right)." src="wikipedia-Low_pass_filter_pliki/d72baa65c67dfb40176258c142991e92.png"></dd>
</dl>
<p>If <span class="texhtml">α = 0.5</span>, then the <span class="texhtml"><i>R</i><i>C</i></span> time constant is equal to the sampling period. If <img class="tex" alt="\alpha \ll 0.5" src="wikipedia-Low_pass_filter_pliki/4366ff02bb229e6ee3d0cef27ee229e6.png">, then <span class="texhtml"><i>R</i><i>C</i></span> is significantly larger than the sampling interval, and <img class="tex" alt="\Delta_T \approx \alpha RC" src="wikipedia-Low_pass_filter_pliki/884081f834d99f1c1f72f98b77a4638c.png">.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Low-pass_filter&amp;action=edit&amp;section=11" title="Edit section: Algorithmic implementation">edit</a>]</span> <span class="mw-headline" id="Algorithmic_implementation">Algorithmic implementation</span></h3>
<p>The filter recurrence relation provides a way to determine the output
 samples in terms of the input samples and the preceding output. The 
following <a href="http://en.wikipedia.org/wiki/Pseudocode">pseudocode</a> algorithm will simulate the effect of a low-pass filter on a series of digital samples:</p>
<pre> // Return RC low-pass filter output samples, given input samples,
 // time interval <i>dt</i>, and time constant <i>RC</i>
 <b>function</b> lowpass(<i>real[0..n]</i> x, <i>real</i> dt, <i>real</i> RC)
   <b>var</b> <i>real[0..n]</i> y
   <b>var</b> <i>real</i> α := dt / (RC + dt)
   y[0] := x[0]
   <b>for</b> i <b>from</b> 1 <b>to</b> n
       y[i] := α * x[i] + (1-α) * y[i-1]
   <b>return</b> y
</pre>
<p>The loop which calculates each of the <span class="texhtml"><i>n</i></span> outputs can be <a href="http://en.wikipedia.org/wiki/Code_refactoring" title="Code refactoring">refactored</a> into the equivalent:</p>
<pre>   <b>for</b> i <b>from</b> 1 <b>to</b> n
       y[i] := y[i-1] + α * (x[i] - y[i-1])
</pre>
<p>That is, the change from one filter output to the next is <a href="http://en.wikipedia.org/wiki/Proportionality_%28mathematics%29" title="Proportionality (mathematics)">proportional</a> to the difference between the previous output and the next input. This <a href="http://en.wikipedia.org/wiki/Exponential_smoothing">exponential smoothing</a> property matches the <a href="http://en.wikipedia.org/wiki/Exponential_function" title="Exponential function">exponential</a> decay seen in the continuous-time system. As expected, as the <a href="http://en.wikipedia.org/wiki/Time_constant">time constant</a> <span class="texhtml"><i>R</i><i>C</i></span> increases, the discrete-time smoothing parameter <span class="texhtml">α</span> decreases, and the output samples <img class="tex" alt="(y_1,y_2,\ldots,y_n)" src="wikipedia-Low_pass_filter_pliki/f0c206e68a41fd151c5671d9772e2dfc.png"> respond more slowly to a change in the input samples <img class="tex" alt="(x_1,x_2,\ldots,x_n)" src="wikipedia-Low_pass_filter_pliki/8b98f712c02e31f319616f5b8bcc05fa.png">&nbsp;– the system will have more <i><a href="http://en.wikipedia.org/wiki/Inertia">inertia</a></i>.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Low-pass_filter&amp;action=edit&amp;section=12" title="Edit section: See also">edit</a>]</span> <span class="mw-headline" id="See_also">See also</span></h2>
<div class="thumb tright">
<div class="thumbinner" style="width:172px;"><a href="http://en.wikipedia.org/wiki/File:Inside_of_a_Boss_Audio_DD3600_Class_D_mono_block_amp.jpg" class="image"><img alt="" src="wikipedia-Low_pass_filter_pliki/170px-Inside_of_a_Boss_Audio_DD3600_Class_D_mono_block_amp.jpg" class="thumbimage" height="128" width="170"></a>
<div class="thumbcaption">
<div class="magnify"><a href="http://en.wikipedia.org/wiki/File:Inside_of_a_Boss_Audio_DD3600_Class_D_mono_block_amp.jpg" class="internal" title="Enlarge"><img src="wikipedia-Low_pass_filter_pliki/magnify-clip.png" alt="" height="11" width="15"></a></div>
A <a href="http://en.wikipedia.org/wiki/Class_D_amplifier" class="mw-redirect" title="Class D amplifier">Class D amplifier</a> with an integral low pass filter, intended for powering <a href="http://en.wikipedia.org/wiki/Subwoofers" class="mw-redirect" title="Subwoofers">subwoofers</a></div>
</div>
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<div class="noprint tright portal" style="border:solid #aaa 1px;margin:0.5em 0 0.5em 0.5em;">
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<tbody><tr>
<td><a href="http://en.wikipedia.org/wiki/File:Nuvola_apps_ksim.png" class="image"><img alt="Nuvola apps ksim.png" src="wikipedia-Low_pass_filter_pliki/28px-Nuvola_apps_ksim.png" height="28" width="28"></a></td>
<td style="padding: 0pt 0.2em;"><i><b><a href="http://en.wikipedia.org/wiki/Portal:Electronics" title="Portal:Electronics">Electronics portal</a></b></i></td>
</tr>
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</div>
<ul>
<li><a href="http://en.wikipedia.org/wiki/Baseband">Baseband</a></li>
<li><a href="http://en.wikipedia.org/wiki/Digital_filter">Digital filter</a>: Another realization of a low-pass filter</li>
<li><a href="http://en.wikipedia.org/wiki/High-pass_filter">High-pass filter</a></li>
<li><a href="http://en.wikipedia.org/wiki/Band-stop_filter">Band-stop filter</a></li>
</ul>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Low-pass_filter&amp;action=edit&amp;section=13" title="Edit section: References">edit</a>]</span> <span class="mw-headline" id="References">References</span></h2>
<div class="reflist" style="list-style-type: decimal;">
<ol class="references">
<li id="cite_note-0"><b><a href="#cite_ref-0">^</a></b> <span class="citation book">Sedra, Adel (1991). <i>Microelectronic Circuits, 3 ed.</i>. Saunders College Publishing. p.&nbsp;60. <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/0-03-051648-X" title="Special:BookSources/0-03-051648-X">0-03-051648-X</a>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Microelectronic+Circuits%2C+3+ed.&amp;rft.aulast=Sedra&amp;rft.aufirst=Adel&amp;rft.au=Sedra%2C%26%2332%3BAdel&amp;rft.date=1991&amp;rft.pages=p.%26nbsp%3B60&amp;rft.pub=Saunders+College+Publishing&amp;rft.isbn=0-03-051648-X&amp;rfr_id=info:sid/en.wikipedia.org:Low-pass_filter"><span style="display: none;">&nbsp;</span></span></li>
<li id="cite_note-1"><b><a href="#cite_ref-1">^</a></b> <a href="http://www.cg.tuwien.ac.at/research/vis/vismed/Windows/MasteringWindows.pdf" class="external text" rel="nofollow">Mastering Windows: Improving Reconstruction</a></li>
</ol>
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<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Low-pass_filter&amp;action=edit&amp;section=14" title="Edit section: External links">edit</a>]</span> <span class="mw-headline" id="External_links">External links</span></h2>
<ul>
<li><a href="http://www.allaboutcircuits.com/vol_2/chpt_8/2.html" class="external text" rel="nofollow">Low-pass filter</a></li>
<li><a href="http://www.st-andrews.ac.uk/%7Ewww_pa/Scots_Guide/experiment/lowpass/lpf.html" class="external text" rel="nofollow">Low Pass Filter java simulator</a></li>
<li><a href="http://www.tedpavlic.com/teaching/osu/ece209/support/circuits_sys_review.pdf" class="external text" rel="nofollow">ECE 209: Review of Circuits as LTI Systems</a>&nbsp;– Short primer on the mathematical analysis of (electrical) LTI systems.</li>
<li><a href="http://www.tedpavlic.com/teaching/osu/ece209/lab3_opamp_FO/lab3_opamp_FO_phase_shift.pdf" class="external text" rel="nofollow">ECE 209: Sources of Phase Shift</a>&nbsp;– Gives an intuitive explanation of the source of phase shift in a low-pass filter. Also verifies simple passive LPF <a href="http://en.wikipedia.org/wiki/Transfer_function">transfer function</a> by means of trigonometric identity.</li>
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<div style="padding:0em 0.25em"><strong class="selflink">Low-pass filter</strong> (LPF) · <a href="http://en.wikipedia.org/wiki/High-pass_filter">High-pass filter</a> (HPF) · <a href="http://en.wikipedia.org/wiki/All-pass_filter">All-pass filter</a> · <a href="http://en.wikipedia.org/wiki/Band-pass_filter">Band-pass filter</a> · <a href="http://en.wikipedia.org/wiki/Band-stop_filter">Band-stop filter</a></div>
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<td class="navbox-abovebelow" style="" colspan="2">See also: <a href="http://en.wikipedia.org/wiki/Electronic_filter">Electronic filter</a></td>
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